The stock file manager should be able to copy files to the SD card without rooting. However, in order to write to the SD card from any other app, you will need to modify your platform.xml. If that doesn't mean anything to you, http://forum.xda-developers.com/show....php?t=1491954.

My microSD card is mounted at extStorages/Sdcard but as read-only that I have to connect to PC to write files to it.

How can I move or copy files form internal sdcard to my microSD card without connecting to PC?

If you just want to transfer file to your tablet, there is an application called MyPhoneExplorer. You have to install it both in your computer and your device. To access your SD card you do need to root your phone.
Connection can be made through WiFI, USB, Bluetooth or Fixed IP address.

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